CHAPTER VIII NUCLEAR REACTIONS A. dotation The nuclear reaction A A(a,p)B + a ■3 + p + <4 is symbolized by alpha, p proton, d deuteron, 3 Pa r t i c ] b s are s jmb o L i z ed by J gamm a ray , a nd f for fission. Q is (+) for an "exothermic" reaction, (-) for " ondo thermic" . ThG thres hold is the minimum energy of the bombarding part- icle in order far the reaction to occur. Threshold is measured in the laboratory . system, and therefore is not necessarily equal in magnitude to Q. If Q is positive, the threshold is, in nrin- ciple, 0. If Q is negative, and If the bombarded particle A Is approximately at rest, then ( see ph. I, rage 5) Threshold energy = (-Q) X Mss_qf incipi e nt par ticle Reduced mass of svstem = (-Q)X^i^- M« VIII. 1 for the reaction symbolised above, B. Gen er al Features of Cross- s ecti ons for Mr clear React ions. The following considerations apply to cross- sections for nuclear reactions in the absence. of resonances. Resonance phen- omena are discussed in section D. Consider the transition A + a— *-B + b + Q, v.here the nuc- leus "A" and the particle "a" become the nucleus "E" and particle "b" Both the initial and final states of the system consist of a pair of unbound particles; therefore the transitions is to one of a continuous distribution of states. The initial state also has a continuous range of possible energies, but the experiment itself specifies a particular Initial energy . There are sl;r.Ilar situations in atomic physics. For example, In emission of a photon by an excited aton, the transition is from a sinr-Ie state to one of a continuum of states: Conservation of energy selects the final state. Another atomic example is the non- radia- tive or Auger transition. An excited atom may have two: possible modes of decay. In addition CCtom, no photon cttom b dnd-a photon to photon emission, the atom may decay by emis- sion of an electron. Suppose, for example, the excitation corresponds to one missing electron in the K shell . The energy made available vahen an electron falls into this hole may be greater than the ionization energy, in which case an electron may be emitted from the atom. Again the final system consists of tv:o unbound particles having a continuous range of possible energies , Returning to the nuclear reaction A + a — *-B + b, we use a general principle of quantum mechanic s to derive some essen- tially statistical results on the variation of the cross- section. From quantum mechanics, the probability per unit time of 141 14 2 transition "Golden Rule No Nuclear Reactions number of transitions per unit time nil W = X 2 «k Gh. VIII w is given by VIII. 2 where H is the matrix element of the perturbation causing the transition, and dn/dE = energy density of final states, counting each degenerate state separately. lXi' a may be the same for all energetically possible final states; more often it defends on the state. (For instance, \H\ may depend on the direction of emission.) Then \H\* in the form- ula is a suitable average over the possible final states .**** dn/dE = oo for a continuum \'Hs\ ^ 0, so that the expression form x oo . This difficulty is box of volume JX of states. But in that case 1M| 2 dn/dE has the indeterminate removed by limiting space to a small but" finite and dn/dE large |M] is then but finite. JL drops out of the result. The number of final states equal 3 the number of states of the emitted particle. Thii is because a change In momentum of one particle compels a change in momentum of the other, by conservation of linear and angular momentum of the system. It was shown in Chanter IV, o. 76 that the number of states available to a free particle, "b" , with momentum between p p + dp, confined to a box of volume JTj , is VIII. 3 and du = This must be multi oiled by the multiplicity in the final state** caused by spin orientation, which is given by the factor (2I b +l)x (2I B +l), where I b Is the spin of the emitted particle and Ig the soln of the nucleus. If b is a photon, (2I b +l) Is put equal to two.*"** dE = v% dp^. (true relativistlcally) VIII .4 where p^ and w, are the momentum and velocity in the center of mass frame of reference of the final (B+b) state. Since B is usually massive compared with "b" , p b and v t can usually be meas ured in the laboratory frame. Combining these two equations: dn = 4-T^'Jl (2%+lX2T B +0 VIII. 5 From this and VIII. 2 we get ho. transitions per unit time LAa|Hl 2 (2V0(2T B + vill.6 Tffi 4 ^ The following equation is essentially section C^v^g per A nucleus: definitior the croso- * Derived in Schiff, Qu antum Mechanics , p. 193. ("Golden Rule No. l" is on oare 148 of this text ) . ** This is discussed in greater detail In section C, this chapter. *** This roint is discussed by Bethe and Placzek, Phys.Rev . J51 450, Appendix, p. 483. Multiplicity is caused by the two possible independent polarizations, -&HBH*- See page 214 for more complete discussion. Ch. VIII Nuclear Reactions No. transitions/sec _ Y .,.-- v *f per "A" nucleus - '** X J "a"rel . to"A" x G A ^B 143 VIII. 7 where A arid. B refer to the (A+a) and (3+b) states respectively, and ru. Is the density of particles "a". Take n^ to "be l-jfe, cm" 3 (one particle in the volume). Then J^ Vjrel.to'A" * a-^-b tt^ /K VI II. 8 Since nucleus "A" is often massive compared to "a", V" a "rel .±o"A" is often nearly equal. tfl'T a in the center of mass frame. 'In any case, these two velocity magnitudes are related by a cons- tant factor. Writing ^"'Vrel .to" A" ~ 1j ~a ' A-?-B TT-K =Wl^>t| 2 A(2V0t2I B ^) n^J h VIII .9 In general, )t is unknown. It has the form J drEfy^ U f in{ ha.l where U is the interaction energy*. If the wave functions used to comoute M are normalized in volume JL , Sh. disappears from the expression \SL H\ in VIII. 9. This is seen as follows: let I have the form, at large distances, K exp(ikz). Then y"|Yi a o.r = Nlfi Setting ifjl = 1, we get N = l/<fcfi? If T, ... -, ard T„ -, now mean the un- normalised Diane wave xnj_Tri.a_L i ma_L functions, the matrix element factor In VIII. 9 becomes sih^ f^T Wl ur lmtal VIII .10 (This may be looked upon as taking Jt= lJ Henceforth we use ~H for JlH . In order to show the meaning of this expression, we write it as pfs£| = [J X Volume of nucleus X ] T mi tial T-fW) VIII. 11 Where H|^r in Is a suitable average of the product of the wave func- tions over the volume of the nucleus. U, and hence the integrand, is zero outside the nucleus. U = average Interaction energy^ depth of potential well. For our purposes here the Important feature of VIII. 11. is Its dependence on the charge of the parti- cipating "particles. If "a", say ; is positively charged, its wave function will be reduced in amplitude at the nucleus by the barrier factor exp(-G- a /2), where, by III. 3, p. 5$. fj^dn. — c barriers VIII. H U & denotes the charge of "a" times the Coulomb potential of "A". Physically this factor represents Coulomb repulsion. The wave function of an outgoing particle at the- nucleus is also reduced by such a barrier factor. The result for the squared matrix ele- ment is: it— \ 2 For neutral particles: IX I <£(u X Vol. of nucleus) VI 1 1. 13 For + charged oartlcles: |Hi*&(U X Vol .)*X exH-G^-Gj,) 144 Nuclear Reactions Ob... VIII femission of negatively charged particles (elect rons)ls treated in Ch, IV ) For endo thermic reactions there is a threshold energy for the bombarding particle. For exothermic reactions in which the energy liberated is much larger than the energy of the bomoarding particle, there are tm simplifications in equation VIII .9= l) the barrier factor exc(-G n ) for the outgo lrig particle is almost constant be- cause it is a function of energy of the emitted particle u , which is almost constant; 2) p^ and v b are almost constant and therefore the statistical weight factor In VIII. 9, Pt/ V a v b* lB proportional to l/T^f These results are now apolied to specific cases to deduce the general features of the vs. energy and d vs. velocity curves. 1) ELASTIC (n,n) (both particles , uncharged) v. . ./^ 'therefore" p|/v v, = C*Wt ) > a conBtant At low energy \H\ Is approximately constant, therefore constant at low energy. ELASTIC (n,n) £. a le^Y e.v. 0f a EKOTHEgMl'G. low energy UNCHARGED bombarding particle, as In (n,.a), ( n , p } , [n , if, ( n , f ) . 4 is usually ,~ Mev «hl le neut- ron energy is— e.v., therefore v b ^ constant . Therefore p b Vv a v b Z lAa • r*l E °C exp ( - G n - %1 . exp (- % } is . & c onstant , since it den ends on the almost constant energy of the out- going particle, or, in the case of an uncharged b , is 1 exactly. Also exp(-G r ) = 1. Therefore a ~ 1/T„ (the "1/v" law) _L EXOTHERMIC oj; 3) INELASTIC (n,n' ) The nucleus Is left in an excited state. The process is enao- thermic and -<J is the excitation energy of the nucleus. 5 or Ircident neutron energies slightly above the threshold, v e=j constant, since the fractional change In incident energy Is small. Put v , changes relatively greatly in this region: v t (C excess o n <c Pn' / T n v n' f energy above the threshold. Therefore ,. v j i oc ^ energy excess 1 . Therefore near the threshold cS CC \| energy excess. INELASTIC iri,v') cr ENDOTHERM1C -T We s hj Id [ ¥ neu TVoN 4 ) ENDO TIIERMI C , CHARGED OUT GO I NG part I c 1 a , as In ( n , a ) , ( n , p ) 3) except t^at the factor exp(-Gv,J Exactly as In case _ operates and is d o m 1 nant . ^ e&pnersy excess 1 X exp( Ch, VIII Inverse Processes 145 5) EXOTHERMIC, CHARGED INCOMING particle, as in (p,n), (a,n), (a.,'3 ), (p, "tf ). For incident energies << Q, the factor p^ /v a T-^ oC l/v a . The barrier factor exp(-G- a ) operates on the incoming particle. EXOTHERMIC <y cC l/v„ s&cpC-Ga) In all of the above, no account has been taken of resonance phenomena . ■ Inve rs e Processes Consider the transition A + a — >- B + b, where "A" and M B I: are nuclei and "a" and "b" are, In general, lighter particles. From equation VIII. 9 the cross- section for this transition is (neglecting spins): , , A _- JffUHr Jt- vt-.,.., i A-*^ 7T*l 4 rt&^b The Inverse reaction Is B + b — > A + a. .2 cr Q SiH k' Its cross section is VIII. 9" |uLr*l is the same In both cases, because the operator of the perturbation is Hermit ian, i.e., I ' : fti*X¥&£\ = \ (w'* U V (tfc\ therefore, JA^-e Cs^-A ft* (neglecting spin) VIII. 14 The same result may be looked at from a different aspect. Suppose we have a box filled with arbitrary numbers of particles "A", "a", "B" , "b" . The transitions A + a ^=t B + b occur. Statistical mechanics asserts that at equilibrium all possible states of the system consistent with the specification of the energy of the system are occupied with equal probability. If a state consisting of a pair of particles ~A + a is called an "A" state, and similarly for "B" state, then the occuoied states in the energy range AE may be divided into the two' tyoes , A and B. Since all states in AE are equally probably occupied, this div- ision Is such that No . occupied A states No, Is- No. possible A states in A E 15" occupied B states No. possible B states in AE The number of possible A states = maximum number of (A + a) pairs times the number of states In A E for one oair - 7? 4"rr fa. _& % At where '( - maximum number of (A + a) pairs formable With the particular numbers of particles cut into the box Initially. Sim- ilarly, the number of possible B states where J} is the same. Therefore 71 146 Ho No. of Inverse Processes of o ccupied A states __ Pi ^b c cue led B states Pfc>a Ch. VIII VIII .16 mw at equilibrium the number of transitions A-^B equals the number of transitions in reverse, per unit time. No. transitions A-^B/sec = (Ko No. transitions B^-A/sec = (No Combining VIII. 16 with VIII. 17, A states occupied ) A ^ g ^ B states occupied )CT e ^ fir to VIII, 17 VIII. 14' iSWB'Uv t^a as "before If the particles have spins, the density of state 3 is in creased If" the spins are I A , I a , I B , I b , the density of A states is increased by the factor (2I A +1 ) (2I a +l ) , and similarly for 3 str-t.pp Thfin t/hft rate of transition A— i-B is proportional to "tes. Then the rate of transition (2I A+ l)(2I a+ l)p a 2 C^ B and -A to <2i 3 +i) (2i- 3+ i) Pb 2 cr e ^ A therefore (2VH2V , <U B = (2^+0(2^0^^ VIII. IS Note that in this formula, O is an average over the various kinds (spin orientations) of A states, and a sum of partial Cf s for various possible final states. * * This may be elucidated by writing O^g more explicitly. Divide C Into contributions Ots) due to various relative orientations of I. and I n . The number of states represented by each relative orientation is 2Sf 1 , where S = resultant angular momentum of part- icles "A" and "a". In this discussion, orbital angular momentum Is neglected. It is Included in a discussion in the appendix of Bethe and Placzek 1 s paper. Phys.Rev . SJL 450. The total number of A states Is (2T A +l) (2I..+I } . The total cross section for transition to any B state is %- E(2 s ^)tf(S) average over spin states. (S takes on 2I a +l values the partial cross if t a <' J i A V "(2T A +i) if"i A <i a .) Nov a\s) section- for various initial values of S, nay be written as a 3um of contributions to various possible final spin states, I.e., C*sl =X ^% , where i denotes a particular final spin state of the B + b system. &&\ contains in addition to the dens- ity of states in ener~.y, the squared matrix element for the parti- cular transition represented by cnSTf . For transitions not con- serving total vector angular momentum , 0(5). 0. For examele , Ch. ¥111 Compound Nucleus Problem to ■ g i Be^ + H 1 147 Design an experiment to detect the inverse reaction" Li u + He" (Design of the aloha particle source will depend on the thresh- old energy for the inverse reaction. From Allison, Shapes ard Smith, Biy a .Rev. 21 550, or from Hornyak and Lauritsen, Rev" Mod. Fry s. SO, 202, we find that Q for the forward react iorTis 2.115 Mev. In the reverse reaction, in order to get 2.115 Mov into the center of mass coordinate system we must give the alpha an energy of about 3-5 Mev, and this is the threshold for the" inverse reaction (See section A). Design of the Li6 target and of the detector, and determining the required aloha beam strength require knowing the cross section. This is got by detail balan- cing arguments from d^^^ , taking into account a spin fac- n° r o° f m 7 ^ - ™ ThiS cr ? ss section is f ound in Livingston and Bethe, '-. rev.Mod.Hiys^ 9 245, p. 310, or in the original source, Allen, -, ^...^u.iny^ y ^-j p. jiu , or in the original s. Ki.ys.Rev ^1 182 Tl937), and is 5 x 10^9 cm 2 " at 0>1 Mev The cross section for the inverse reaction Increases raoidly 'as^the volume of phase space available to the proton is increased therefore it Is advantageous to use alpha energies an Mev or more above the threshold of 3.5 Mev. Higher energy orotons also penetrate the Coulomb barrier readily, and are easier to detect A qualitative curve of cross- section for the forward reaction as^unction g^^is given in Hornyak and lauritsen, Rev. D. The Compound Nucleus ""in the diagrams of section B it was assumed the lH^was approximately constant, except for the Coulomb barrier factor Often, perhaps in most cases, the matrix element has irregular variations. This phenomenon is called resonance . For ^xamol- if [n J S J Process 1» indium, there is an extremely pronounced peak in rf at a neutron energy of 1.44 e.v. d reaches 27,000 barns at this energy. (one barn is in '' '" ' " this resonance peak is 0.042 e.v, = p curve of rf vs. energy has the form is the resonance at E R - 5.2 e.v. rf for the (n,# ) reaction in silver. In this case d reaches 24,000 barns, and the peak has a half- -24 em .; Near l7(E-E R } 2 . The half-width of the resonance, the Another example will H - 0.063 e.v consider the reaction n for a, 0; assume for A, initial spin states : tial spin s bates for = 4; for S B 3, The 3/2. + A — >C + ±; and for _. ^2(l)+l)(S(l/2)+l) = 6. total angular momentum 3 1/2, (2(l/2}+l) =2. *)s = i < *%-f ) + Now Hie first term represents states having S = 3/g. For a spin3 are, for n, 1/2; The total number of The number of ini- = 3/2 is (2(;5/2}+lJ ^Ai% transitions to any of the final spin given initial orientation, there is only one. Similarly, the second term represents transitions to any Imal state having total angular momentum 1/2. hut dnce the spin of the a = there are none, so 0T(&) =* 0. Mien orbital ang- ular momentum is involved, there may be more than one way in which tne given Initial state can form a final state, so that &&) for tf^lll' I 3 a T °^er the various possibilities. See BethTand Plaezek, Pays. Rev^ bl 450, appendix. 148 Compound Nucleus Ch. VIII The explanation of this phenomenon lg baaed on the assump- tion that the transition A + a— hB + b occurs through an inter- mediate state C: A + a — *- C —- 3 + b State C Is the "Compound nucleus". The idea of the compound, nuc- leus is due to Bohr.* The idea of how resonances in cross section result from this assumption can be obtained from the quantum mechanics of second order tnansitions. The probability of transition, per unit time, is given by "Golden Rule No . l" :** trans, urob./sec — 4\ ^*X BC 2 /energy ■ x density or \staies VIII. 19 provided there are no direct transitions from A section is, from VIII. 9, , ' ...:-K which becomes , analogously, &n\ i v*v b 1 HcA^cl'-^ "A^B-t^ I Ea-^ % v b :o B , The cross VIII .9' VIII. 20 Near E^ = E c , (resonance), tS is large. This formula gives infin- ite <5 at the resonance energy, hut the formula doeH not take into account the short lifetime of the compound state. A correct formula is derived In section F. The life -time of the compound state is long enough for the nucleus C to "forget" how It was formed ,***and this results in a basic simplification in the interpretation. From the Heisenberg relation AtAE & }4 , the lifetime of the compound nucleus and the uncertainty y in Its energy are related P > JL lifetime VIII. 21 The reasons why the compound nucleus has a. lifetime greater than zero are the following: 1) For charged particle decay, the barrier factor (VIII.12) reduces the rate of decay, 2) Decay by X radiation is very slow compared to the times in wnich the nucleus changes Its organization: the lifetime against V emission is ~ 10~13 - 10*3.4 sec. The characteristic time of the nucleus, I.e., the time for a iiucleon to cross the nucleus, is —(size)/(velocity) vug 10-2.3/109, or about 1CT 22 sec. 3) A particularly important reason is the tendency toward equlpartltion of energy in the nucleus. The excess energy due to the absorption of the bombarding particle is distributed among all the nucleona. It Is rare that there is a fluctuation in which a large fraction of the excess energy is on one nucleoli. 4) Selection rules forbid some modes "of decay. * Bohr, Nature 137 344 (1936) ** Schiff, n. 1967 eq. (29.20) a-ture 137 . P- 1967 **'* Discussed in Peierl's review article In Reports on the Progress In Physios VIII (1940), Fhys. Soc . of London, 1941. C ompound ETucl eu B 149 c- Ln- tie ted 1 r ) s to :res3 Ch. VIII S. Examrle of an Unstable Nucle us " An examnl c of a nucleus which plays the role of an intermed- iate-state compound- nucleus for several well known nuclear reac- tions Is Be°. The ground state Be decays Be^ 2 He as follows: ■* 110 Xey r » The reaction is barely exothermic. The G9i*W exponent for decay into a's is low due to low nuclear charge, see equation VIII. 57, o. \62 . The theoretical estimate of the lifetime Is ICT^sec".** , corresponding to a width of between 1 and TOO e.v. This time is long compared with the nuclear characteristic time of 10" 22 B ec . J hence the width of the level is small. Information on the excited levels of Ee S can be obtained from studv of those nuclear reactions for which Beo is the inter- mediate compound nucleus state, such as Li' (p , ~y )Be e — j^2a, Li7{p J n)Be7. These reactions are discussed here. The energy levels are plotted in FIG. VIII. 1. l) a-a scattering. For two Coulomb centers, the total scat- tering cross section is oo . ¥e may study the scattering at some anr-le'"not near (90° in center of mass system is best). We expect peaks in the value of CS when the incident relative energy equals the energy of excitation of an excited state. For a s scattered on a's", the first such resonance should come at 0.110 Mev (in center of mass system), corresponding to the Be° ground state. This resonance is presumably Ten sharp, a few e.v. wide, as mentioned above. It has never been observed experimentally. Problem, Discuss the possibility of experimentally observing the resonance expected in alpha-helium scattering at an energy corresponding to the Be 6 ground state, i.e., 0.110 Mev In the center of mass frame. (The Coulomb barrier keeps alphas of this energy at least 5 x 10" 12 cm apart classically, so the effect of nuclear forces is probably undetectable. Also the experiment is difficult because the range of 200 Kev alohas is so short that it is hard to shoot them through an appreciable number of scattering centers and detect them. Any' attempt to detect a resonance might- be guided "by the experimental procedure of Devons ( Proc .Roy .Soc . A 172 127 and 559 (1939)), who Investigated alpha-helium scattering at higher energies. The theory of a-a scattering and its rela- tion to the Be^ nucleus is given In Wheeler, Phys.Rev. 5j? 16 and 27, (l94l) J A second resonance, this one experimentally observed, Is at ^3 Kev. The barrier factor is lower at 3 Mev, hence the state has shorter lifetime and greater width. The half- width is estimated to be 0.8 Mev. Further resonances In a-a scattering are so hroac as to be scarcely recognizable as resonances. All the resonances men- tioned so far" correspond to states of even parity . This is be- cause a's obey Bose-Einstein statistics and have symmetric wave Hemmendinger; auoted in Seaborg and Perlman table Rev. Mo d . Phys ^ 20 585- Wheeler, Phy s . Rev . 59 27- Isotopes , Compound Nucleus 150 Compound Nucleus ch . VIII functions , * The incident a ' s Mil hav e ango lar rsomentuiB 024 with respect to a target a particle. Therefore states of Be* detectable by a sea Iter Inc. IjS helium are even states. Not all states of Be 8 are ever;. Odd states of Be 8 c-mot decay directly into two a's or into the even Be 8 states mentioned Emission- of electromagnetic radiation must occur first, " : - i -- -'■ "hange to an ®J| n state by "mech- Char.ge of parity above , because an even state c ™t anlcal" , i.e., r:on- radiative, interactions occurs in emission of photons. 2) bl (p,K)Be . There is a nrominent and narrow resonance at a proton enerrrj of 440 Kev . This Indicates that the life- time of the excited Be is long and thus that it is an odd state. It decays through the relatively slow process of ~W emission to the much lower even Be states. The energy of a Ll 7 and a proton separated and at rest is 17.2 Mev higher than that of the ground state of Be 8 . Tbe Jf 's given off in decay from the excited Be b state produced in the IA?fot, ^ ) neaction hav e ener gi e s of 17.5 Mev and as. 14 . 5 Mev , indi eating tf decoy to uhe two even states mentioned in the oaragraph on a- a scatter- ing. That the excited Be 8 state oroduced in Ll?(p V) is oM accords with the follov.ring considerations. The most probable ease is for the bombarding proton to be in a high S state which is even. The hV is odd, as is suggested by the arguments in the following paragraph. Then Li 7 (odd) + oroton (S state) is an odd Be state. The picture of a nucleus as built up of "shells" of ^rotors and neutrons, somewhat like atomic electron shells, su.g-ests that Id' in the ground state is odd. Suppose the average potential for the nucleors Is a square •jell The single particle approximate quantum mechanical solut- ion to the problem loads to orbits which ^ay be designated' Is lp. Id, etc The Is orbit accomodates 2 neutrons (soins opoosed) and two protons (spins opposed); ip accomodates 6 neutrons ana 6 pi'Otons , etc. ? yH would have the configuration; protons: Is 2 lpl neutrons : 1 s 2 ipp o, o or a total configuration: (l8 2 lp; ls 2 lr 2 ) , which has ±\l\=-% and is nence an odd state. This model of the nucleus is discussed in _ / change s '■does not x -^-x,y ■5'3tl IS Chan; y ,Zr-V - z, then the ? * For two identical particles, parity of the state and symmetry functions are simply related: If the wave function"' ign when space is inverted by the operation ias( odd oaritv. If ? chan Ges ^even ' does not change hen the two particles are interchanged in eosition, then T is antisymmetrical symmetrical Operation of inversion of space: Y(r-,,r ) — *■ t 1 (-n ,-rvJ Operation of particle interchange :f r!'r,) — *- + flro «T But for identical particles, r-^ i^rfL ^ s^fi ^ £lf-^J - T(r 2 ,r-j), and inversion is equivalent to particle interchange. CH.YITI Be Com|=ound Nucleus 151 Be" tU IB. 5+ 7 pwi -oIlO Zoo 19. ',J IS. 1 3 17.6 I in ' ' i". 2. In R 8 be pis.Ynr. 4)0 *) 2.22 Cu ft e s give cross section for the partic- ular reaction. Numbers l), etc. refer to subsections of section E . (Based on "Energy Levels of Light HucleV' bj Hornyak and Lauritsen. Rev .Mod .Phys. 20, 191 ) O of energy (Energy in Me v.) Be c 152 greater detail in section 7* Ch. VIII Compound Nucleus in section K. 3) Lt 7 (p»p' )Li 7 *. This is similar to 2) except that a r rot on is emitted having less energy "than the incident proton, leaving Li? in an excited state. The resonance in C is observed at a oro'ton energy of —1.05 Mev . Li'* decays by emit tine a ° of "bout 0.45 Kev. The 0.45 Mev splitting bet ■-.-sen this excitea state and the -round state Li? may be due to enermy difference between Pl /n and py states of Ll 7 , on the nuclear shell model: SS.T3 Mev. <■ U t7 17.21 ,45"3Mev X Be* Cem^»>»4 Nucleus ^r^^Msta*! tsf eder-g/^ \ Mev Li 7 + p(l Key) — Be — 4 lf£nevg/ = O at, of Be s , gvoun d stat Li + Li V + P' 4) Li 7 (p,a)He > 7 i md odd, the incident nroton must be in an odd. state, state with respect to the Li 7 nucleus. Ko reson- Since two a's are in an even state, since Li 1 probably a p' ... ahc'es are observed. None is to be expected, since all even resonances are extremely broad. Note that all the observed re sonance s 7 are odd^level s , ,„__,+. __ D .^„„ r .„ 5) Li'(o,n)Bn'. A resonance is observed at a proton energy of— 2.22 Mev, corresponding to an odd Be b state 19.15 Mev above the ground state . ,- 6) Li 6 (d,a)He 4 No resonances. Evidently Li D + & form an even, state, and quickly decay to two a's. Problem: Design an experiment to observe the famous 440 Key resonance in the reaction Li? + p— ^Be" -+/+ Be 1 - ->2a ■ (This exoeriment has beer performed by ha Ike r and McDaniel^ using a gamma ray spectrometer which measures the energy of pair's orodueed by the gamma ray ( Phys.Rev . ^4 515) and by Delsasso, Fowler" and Laurit sen ( Phys.Rev. 5JL (1937)) using a cloud chamber to detect the gammas by means of electrons pro- duced in the chamber by pair production and Gompton collisions. Recent electrostatic accelerators have been equipped with electrostatic velocity selectors which provide an energy spread in the motor, beam of less than 300 e.v. at 1 Mev. In order to take advantage of this narrow energy range, very thin targets must be used. These oroblems are discussed in 'Gamma-Radiation from Excited States of Light Nuclei" by Fooler, LaOTitaen and Lauritsen, Rev. Mod. Hays. 20 236 (1948)). F. Quantitative Development of Resonance Theor y; Brelt-Y.'i.-.ner Formula . t ^ \ In this discussion we use as an example the In, ) reaction, 1 . c . , rad 1 a t i v e c a otu r e of neu fc.rone , wh i ch is a n 1 mpo rt a nt re;:, c - tion. As in the preceding sections, the energy levels of the ini- tial and final states form a continuous distribution. The exoer- ment picks out the particular initial state. ^y ie resonance phenomenon that we wish to describe is very enerry -sensitive. Ve shall attribute it to the existence of: compound rue "Lens state C, connecting with the Initial and fir states by matrix elements "H Bt . and Mfat , where the notation is 'i-nn.l e Bj Ch. VIII Resonance Theory 155 given in FIG. VIII .2 We shall neglect matrix elements of the form H & q> j Mty&tiA M a y , which connect the initial state with itself, the final state with itself, and the initial state with the final state without the intermediate G state; i.e., no direct transitions from A to B, STM"E A CornswJD Nucleus C STATE B A+/n C B *t inde* a. — — index b (discrete) (fort-hrwous} PIG. VI II. 2 The problem of computing transition probabilities can be done by using the complete machinery of time dependent pertur- bation theory-. In summary, if u* are the time independent elgen- functlons of the unperturbed states having energies E. , the true eigenfunction for the perturbed state is Y =1 % A, J, E,t VIII. 22 where a 1 are the amplitudes of the unperturbed states In this expansion. For the method to be useful, the true eigenfunctlon must not differ greatly from one of the unperturbed eigenf unctions . If there Is no perturbation, a n = for all n. With the oertur- bation,^ , the a's change according to the equation a = ^Z H^ cue s(w^* K -A* Hjt^dx. VIII. 23 If j just before the perturbation is applied, the svstem Is in a state represented by uv, i.e., a, = t- k , then some 'time after- ward there Is some probability or finding the svstem in states other than the k zh . For states differing in energy greatly from state nnm.ber k, the exponential in the above equation oscillates rapidly and the change In a ji tends to average to 6. to the , of B Applied states by eL E„ by d c (see state Is near enough to be important the one- level Breit-V.'igner formula.) the amplitudes are " for present problem, we denote amplitudes of A states by a-v, , and of the compound nucleus at rd ■:/'- ill. 2) . Assume that only one intermediate (The result is theft called The differential equations a. = "^K q, = -^H tc e ife-Ee)t Sx 1 a. fc-t The lnlt: _al A state , say ct n = 1, a„ = for a £ a J* a, -^ZHcbe M^-Ztf Is b chO£ en exoerlment . VIII. 24 At t It I m, Ch. VIII Resonance Theory 154 - ■ n h,rt a ^ a increases, representing At t = a a = a. = but a ^ O .a intermediate the build up of probability that tne eys airplitu des a c and State C. When a c becomes larger tnan , the *»pJ-l de £ ot9B a b begin to Increase. Tne a^ 1 *^^"*^ a^ituds that A + ^ ri^ous ,ay to solve -r the transition P^abllltles * is to carry through the solution of the system oi ^^ollO^E 1. a practical way to K et the answer quickly. Assume for the sa.e of «*^^ t t*^ ^tn^ compound nucleus state is ^^^^^^Son^rooabillties are tegrate either to A or to B. The transit^ P ^ ^ ^ ^Sa b Ii;SiS B o?S^ta?e SSst'a certain mode of decay. Probability of transition to B - ^ - £ [Ate (per sec . ) = i4^4 c3 VIII. 25 where we have put ^/c = p y , v y - B ■ Probability of transition to A = ^r -^l^vi * (oer sec . ) (Trc Wave functions are realised in a volume Jl = 1.) The decay from the state C ivould folio,; the eauation; Prob. of occuoation of state G - g i'(t^) VIII. 27 Actually the probability of occupation of a states ^am P litude| 2 . .If we ignore the phase factor, _vrejwrite_ amplitude = NJprobability t/ I ±A_\ or satment 5 4 — ( VIII. 28 Kere thorou^ treatment confirms the result of this step. Defining VIII. 29 VI 1 1. 30 n ^P + P -fe/a) times the total probability of ^structlon I rrU+ U -^% % ^l intermediate state, per unit tune. ( r is sometimes defined ^h^T^T^ZlA^ ce^ritSn probability of decay p er uoit -.lme.J me.- vjl-l-u- __ . * Breit and W^r, BHfeBg* g g9 Clg6) ** Froir now on, Q c is not ae^essaxxiv e.quaj Ch, VIII Re s o na rxe Th eo r v ' v 155 We muBt add a tenm to account lor the accretion J*., e - —*■ ""** s* l.^-x:u uu account lor the pncrp+in of the state G. State C is IiM frcm a+ .^ . h ' „ C f^f appro-n^iate +pt«, fv,™ -,-. t L IT irom state A accordin- to the ViilJJ, rametf £* " S ati0r ;r e °^ **»*«* to equation Therefore we try VIII. 31 SmK^SitV^S S^TS^f eqUati ° n 3atlsf nn G the initial a, == ^H^te^-^-e* 1 ? VIII. 32 tJsl nr fho ,,-„ ■*-•„■ experimental wid?L! h t\rSiSS£? ?i^ C Sr S t^ °^ /tol ^e Of the energy turns out usually to be ?esf ThS 1 gSfg ™ 3 St f e G ' ^ ■ „ seconds the term exr>( Vi /%) IT ^ C * Arter ^™ i^ 13 Jn e;x i'\-\ V)iJ is nearly 0. Therefore VIII. 33 P'+fec-EO' ri2. 8t Sen^ lldS ^ ^ ^ am ° Unt re3?rea9n ted b ? **• «1« of Number of reactions to the ri^t/sec B + y } = Ucl "^r Number of reactions to th= L6£t/fie* (A + n^G-^A + n 1 ) " = L X VIII. 34 T' LTi, examine" k] ^"it^SarS^^T^, ^ ln the flml fo ™> « c „ J ' cl early Ilas the form of a resonance curv^- By definition, T aaJ T are fcrj* ■•'-..-. connected by 2^ = ^. Slne:# " I I turns out to be approx- imately the uncertainty* in ener G y of the state, this "is an aspect of the Heisenber- uncertainty relation AETS'tf. "Energy /v.dt h eft %_ maximum ^ O f -1 Returnin G to equations VIII .34, ;,rit!n. ; them explicit! Wo. ncactions/seo foWl* A^G^E (per = fT~ . — unit beam density) ' ^v t e~-£«J No. reactions/sec |>t co '* '^'-^^ unTt bTam density) ^'+ (£t- E g J a fcjj 7 ~ *W)^<ft Viii. 35 where the definition of Cf 1 r equation VTTT 7 i„ assume the r,robl«n jq r„,™*n , q \ , lon ' m '7 is used, and we cf neutrons"= lT -iodized to volume^ = 1 and density 156 Resonance Theory Ch. VIII (in deriving this result more carefully, one should substitute a c into expressions for a a and a, , then put these g, ' s and a^' s into the correction expression f on a , and check that the result is consistent.") , ,-; J _ ^ ] ^ Ca > { M ^j p, + { ^ T VIII. 36 21; i>u„r The next step is to replace \%J with more useful parameters. We use equation VIII. 26 which nay he written M 2 "" ""'" m« * = i¥?T 3 1 ^ ! v H VIII. 26' :f | ra denotes U for v„ = velocity of neutron if it had the right energy to hit the resonance exactly = v R , then ~i^)H*J e M' VIII .37 provided 1Hcq,| varies little with change In velocity. Rewriting VIII. 36 in terms of the parameters v R and ["*- , °fm,*l "- C%/n) — 2g 1 2ir^rJ VIII. 38 VIII. 39 M**S Wow use the relation ^ =i U, K Cv/v R ) , where v is, as always, the actual velocity of the neutron. Also,^ ^ tf/Mv , | ^ ^ /Mv R . M = mass of nou.tron . rvr R VIII. 41 v = actual velocity, v R = velocity of neutron to hit resonance exactly. .The aensral nature of VI I I. 40 and VI I I. 41 is shown in FIG. VIII. 3. Note that for energies "below the resonance , the eqvntioas above behave as described in section B, p. (44, So far, angular mom°ntnm multiplicity has been neglected. For the .{v., IS ) reaction the spins are: feutron, I H = l/2 Initial Nucleus, I A Compound State Nucleus, I c -' J C^ih/h) F1G.VIU.3 CJli VI11 Resonance Theory Final Nucleus, I B Photon of fixed momentum has two degrees ef freedom and ia lifce a uarticle of sein l/2 (see ref., footnote *#», p,|^j ' ---^j* Sl °'" r r" eutr,0n tactions the neutron usually has no on-it-i DJmTl X G - % i 1/2 resonate. The cross sec- tion must be multiplied by the probability that, -or giver, I. am are^Tf^^/S ^ ' *** "^^ ^ the ri " ht ***»t*tl<m. There - T Wo V + * initlal St " tSS havl ^ ; t0tal an C^r momentum - I A ^ + 1/2, ( 3pln . a parallel), and 2(l A - i/o ) + x st ,, tR3 ^ total arxcular momentum = I A - i/ 2 , ( api , s ant inarain el ) . Ihfi "" total number of states Is a<l + i/ 2 } +1 + 2(l A -i) + i = 4l + ^ The probability that an ineomlm neutw™ ,diT ^ f ■ state uith the nuclei-. j^ mlIJ S n "' ltl011 ■'HI f™ a scin-parallel 4r a + 2 21, + 1 VI II. 42 and that the neutron forms a state with spins anti-parallel is j^I A -/ a ) + [ _ T A 4I A + Z 23* + VI II. 43 V: ' '' " - l be ^oniblned, ~ivin S for the final equations; VI 1 1. 44 ^)-^U,-^^s Cfyyft,} — 4-TT A K £J& *r v s +^y 21* + VIII. 45 where the + is used if I Q - r^ + w £ - ar)d _ There is little or no resonance if b i t neutrons. C F ^ if I + 1/2 , for - 1/2 slow nomena in mo-°o 1 S 4r + 10n CU ^ eS r ° r £ ey -*^ ^bsorntior and scatWrinr il . , "Science and Eiipi- m^j^ 19259 fl.947) or in Goodm neerm;;; of luelear Power", Vol. I. ^ m „ At the ^ddle of the periodic table, A = i 00 to 1 50 fn ¥ 1 aroc esses are ^rmil npn+ n-p+ D v, «„ -. " -o<J , in, C ) for /Sta is ~ fSr tL^ pGrl f iC f^' the M ^0 energy Shis amount At +m * h ' L compound, nucleus is excited by about closed s^c^+niL ? r r °f ^^ation, the levels must" be VIII 4 3 ht t ° n " U ? 5? r ^ c -- 0Sel 7 spaced resonances, FIG viii ^. But at Iovj excitations vre find that the ener-r levels ***** Cr ° 3S ^^^^^^^^u^Ti^rcso process^ 158 Experimental Resonance Data Ch. VIII are of the order of 0.1 Mev aoart.* Evidently the— Is a -reat S^ Se ^^ 1SVel dens "y wit* increase in excitation onei {^y . The increase In level density has been lucidl^ eXDlaiaed by Boar. Suppose that the nucleus resembles somewhat a collection of narmonic oscillators of different frequencies «* . Eacl has energy gc**^ (ignoring zero point energy ). The total energy is E =1W^ W ^*-~^u^ . For low OMltatior _ f fBW Qf the Kj differ from 0, and the energy channea in iumns of ~&*J which we may take as -v, 0.1 Kev . For large excitation, say 10? e!v many rjj s are largo; there are in general manv sets of m*a that £l> a J^S 1 e r- Gr ^ near t0 a lar Ce value of excitation Each such set of n t 's will in general giy. e a slightly different total energy, therefore the density of levels will be large. This result clearly depends on having many nucleons. Ve expect few energy levels and few low energy resonances in the light elements the number of resonances Increasing with A. This expectation, is fnl- lll.lea to some extent, because there are few loiv'e-iem (<1000 e . v . ) r e s o na n c e s In el erne nt s lighter than mangane s e . ^ ' " + n ( F S r , vo ^ hi #i atomic number, the density of levels accessible to [n^ ) processes does not increase; altho hi~h«r A means ior P degrees of freedom and denser levels, «**«« more the binding energy of the neutron decreases from ~ 8 Kev to 5 or 6 Mev, and thus the states for for- mation of compound nuclei by absorp- tion of a neutron are at lower excitation energy and the density of levels is correspondingly less. These opposing trends tend to can- cel . Not all levels are detectable by (r.,^ ) processes. Those for which the spin of the compound nucleus state is not equal to ■^A ~ 1//2 ' a "l/oi" parity Is not conserved, do not give rise to res- onances in { n> X ) cross section. Throughout the periodic table I Y is roughly constant at a value 0.1 e.v. v.dth variation by a factor of 10 either way. For medium and heavy elements, V n s= lcr 3 ponds to a life time against neutron decay oi ■ Is large compared to the nuclear oeriods of about HT"**- sec- the captured neutron moves around — Iff? times before a neutron escsnes ihis very low probability of escape can be understood by consider-" lng partition of energy among the many nucleons. (See n". 14 & ) this process to make neutron emission slow operates bettor as- A becomes larger. In light nuclei, VL. is larger than in heavy nuclei, as ore expects from the partition of energy argument. " \Z ler-e means large probability of scattering comoared to cantnre. In mangan- ese^ for example, the resonance at 200 e.v. is" almost pure scat- Energy Levels of Compound Nucleus FIG. VIII. 4 c o rre s- e . v . Shi i 10~ J - seconds, this -21 * Known from gamma emission spectra and from comolex alpha spectna. lM e v c; the escapes , orsfder- }. ) Thl s becomes Ch. VIII Nuclear Gas Model 159 out. 1% as larpe as the VI I I. 44 teri-'i" * Trio rjeslc in absorption is only . scattering peak. The Breit-Ki -ner formulas VI II. 44 ano 05 are fitted to this resonance by putting b> &*1 o.v. ana V^ 10 B.-.V.. In TP rv UPht elements, there are no (n, 5 } re sor-.ar.ee s at ell. There are, however, (n,p) and (n,a) resonances. The Breit-Wi-ner theory may be extended to reactions involv- ifin clia^fed rarticles by ircludinp the barrier factor clue bo Coulomb" forces.** For encrpies above the hei-ht of the Coulomb barrier resonances are observed. For example, the reaction ''■' ^ H? has a 'uroad resonance at 0.27 Kev. Li 6 + r_ ■He Problem. Assuming that U»\y, find the possible values for C for the 280 e.v. resonar.ee in manganese. . CJ - , , \ ( i at 280 e.v. is 2.73 x 10" -1 cm. The spin of hn JJ is 5/2 J Froi? enuation VIII. 45 f At resonance k = 0. Then ipnorinp Vy compared to k , we pet If v;c hnov: that P^lOO hV , ve can also find %,19. 5 Stati sti cal G-as Model of the Nucleus Va r i ou s m od el s o f the nu c 1 eu s empiia size various^ different features of the nucleus. Me si nple simple f" -}} nuclear properties. V;e shall consider tne statistical oi ^f model, then' the liquid drop model apolied to ussion, anc finally the nuclear shell model,*** Tne -as nodel nic tares the nucleus as a C as of protons and neutrons/' This model ignore, surface effects Itte na .piliaritv, a serious omission. The volume of the pas Is W ,5 ; "^ Due to this restriction to a small vo u,e ^^ILsTv^r/Sfih mm for s rertide has widely spaced levels.*--" Dffles.s very p^gn e^ciStlen^nercies are postulated , P^I^a will ? c W toel jx- _.« + w911 »i,i 6 states. For the usual nuclear excitation encr.ieb oflQ lev or so! the nucleon ras Is almost completely aepenerate . We shall use the m model to compute an approximate nuclear .etentiarSii'depth, !o explain semi^antitatlvely the increase in nuclear level density uitn ener G y, and to consider smisBltti particles as an evaporation process. The excitation of the nucleus, the ^tert to .chich hipiher T - cerr-sS? to complete degeneracy . At T = tne number L-%?aKa ri to the hiphest one occupied just equals tne nun ner or reticles, either Z or A-Z, depending on uhether the proton or neutron part of the &s is in question. 71 6 '.;. * Seidl, Karris and Lanpsdorf, Phvs.Rev #* See Bethc- B, Ch. XIII, pape 186 . „ fl ... fa1/1 *#* Various nuclear models are discussed by L. Rosonleiu, Nuclear Forces I I, n. 185 , and by Seine B, p. (J **** Equation VIII -5, "for enerpy density of states, cortains the volume . 160 Nuclear Gas Model Ch. VIII The number, n, of states of momentum less than r> ,„ of a proton confined to a voiumeJQ, is r ' = 2 ■ — ; rvj (factor 2 is for snln, see p. 142} 3(2ti40 3 At complete degeneracy n — Z, therefore ^bX^A * f %Z ■-■■ (? v )^k& 4 VIII. 46 Similarly VIII ,47 In the crude approximation, number of neutrons — number of protons =N=A/2, 4L - (3Tl) * inm A = 1.05" x l& cgs umts 0.5*^A 7 3 ' : VIII < 48 ;S!: = :S independent of A. The corresponding kinetic energy is «v 21 Mev ^y^a . ^Mf, Is the kinetic energy of the highest occupied neutron state. This energy is measured from the bottom of the potential well. Further more,yW is about 8 Mev below 0, a 3 shown in FIG-. VIII. 5 This fixes the depth of the well for neutrons. Actually there are fewer protons than neutrons in the nucleus, but the topmost proton level must have energy^^o , otherwise there would be 3 decay. Therefore the depth of the well for protons is somewhat less than for neutrons, by this model. Also, the Coulomb potential acts on protons . Ipl-oton |ootenilQ I v. Oeotron potential 8 Mev ~ 21 Mev B&."ES,S" or The following, is a calculation,,! corresponding to a usual value of nuclear excitation. From the statistical mechanics of a degenerate Fermi-Dirac neutron gas.* the total energy of excita- tion (measured above the T = level) is, for neutrons (FIG. VIU.&) # Mayer and Mayer., Statistical Mechanics , p. 374. The total energy measured from the of kinetic energy is 1 ; + ""J VIII. 5P VIII. 49 E*-*»*xL'+1F(jS3"+ <m. viii Nuclear Gas Model 161 How put In Tor / M 6 = p^ ax /2M , and then for p put VI II. 47. The result is Similarly x-: ftfejA h&^s rV^^^-^^f) 2 172/ # _n- fc. p = fW ^ f^f f/rf! VIII. 51 VIII. 52 where ^ =nuclear volume, T = nuclear temperature, and M = mass of a nucleon. The total (neutron plus proton) excitation energy. / FIG. VIII. 6 T E t - E n + E P' is X, (Me/) VIII. 53 Where E^ and (kT) are in Mev. If A = lOO^and Z - 44 , HL = 11 (kT) 2 . Then (kT) = 1. Kev corresponds to Ex— 11 Mev. If we attribute to each kT a "decree of freedom" in analogy to classical statistics, then in this case there are only 11 decrees of freedom, whereas there would be 3A degrees of freedom in the classical E£ s . The increase of energy level density with energy of excitation can be computed using the statistical-mechanical definition of entropy: Entropy = S(T) = k(ln w(T) - In w(0)) VIII. 54 where w is the total number of quantum states available to the system at the specified temperature. Fro is thermodynamics, str)= f T ^ » r T 2aVTdT = Zo ^J sine* E-a^feT}' VIII. 55 a a So 3 =^ ^fl^^ r From the definition of entropy, ,7%. _ _ US (Tj _ dens ity of states at temo, uj ( 0) ~ density of states at temp. e VIII. 56 e = e 11 (kT)^, For A = 100 and Z — 44 (same example as above), E and at E ~ Mev, the excitation of a compound nucleus after capturing'?" neutron, e ^ ^ e l9 ~ po 8 . Therefore, if near excita- tion the level spacing, is ~ 100 Kev , at 8 Mev excitation it is ~ 1 millivolt, by'this'very approximate calculation. This spac- ing is probably much too small, although one expects this calcu- lation to give' a greater level density than observed by resonances because resonance experiments detect only levels with spin compat- ijgg Nuclear Evaporation ch ■ "Hi ible with those of the initial particles. Nuclear evaporation . The emission of a neutron from a nucleus may bo considered an evaporation of a particle from a statistical group of particles held in a potential well.* In such an evapor- ation, the particle carries away an energy of order of magnitude ST, which is, in general, much less than the total excitation energy of the group of particles.**-, For example, suppose a nuc- leus'" of A = 100, Z"= 44 has excitation energy, £=20 Mev. The bind- ing energy for one neutron is about 8 Mev. The temperature will be about, ""l. 3 Mev. A neutron will, on the average, have kinetic energy of about 2.6 Mev after escaping. Therefore the nucleus is left With an excitation of about 10 Mev., which is sufficient to emit another neutron. After the excitation is reduced below the binding energy of a neutron, the nucleus may decay dj gamma omi-s- si on. Protons encounter the Coulomb barrier, which has a height of ^j .gr-10 Mev for medium weight elements. The probability that 2 + ~ 5 Mev is concentrated on one proton is small if the total ex- citation energy is of the order of 20 Mev, therefore protons are less likely to' escape. At small Z, the Coulomb barrier is rela- tivelv small, and p emission processes compete with n emission processes. For largo Z, n emission dominates. When the energy delivered to a nucleus is very than the binding energy of a particle, as in nuclei c o sm i c ray part 1 c 1 e s , a ' s a nd ev en larger nu cl ea r "evaporated". In photographic emulsion or in a cloud chamber. a "star", having 3 to — 20 or so prongs, is observed r (See much larger excited by frarj-ients are 17?) Problem Plot the probable number of neutrons that an excited nucleus will emit, as a function of excitation energy. The plot should take the form: probable Plot up to energies such that four neutrons have probably evaporated . Discuss the probability for emission of a proton at this excitation, nurvber <£ neutrons Emitted Assume A = 130, Z - 54, and that the binding energy of a ncutron or of a proton is fl Mev . (The temperature of the nucleus changes only a few percent when a neutron leaves, so the error is not great in assum'ng that T is constant. Gamma decay may be assumed negligible for energies of excitation "of over a few Kgv above 8 Mev (see Bethe B, p. 160). Considering emission of mavtnons first: For the energy region 0-8 Mev, no neutron can lie emitter]. For the region {3+n, f sv: Kev)to 16 Met'-, one neutron is emitted-. For 16-24 Mev there are pb.mpeting modes of decay, namely, l) one neutron may take cnoumh energv'to prohibit further evaporation, 2) two neut- state population met ion for the nuc- Energy, E b ro"-s may leave. The lcar mas has the f orm : We may approximate the tail of this function b an ezepo .ential, and say that t.-~e A rrobabillty for escape = C-v/u.-tt'e'^' 1 '; £>Efe where C is a norrmlisatlo:: factor and £. is energy measured atove^, , the Fermi energy, th e Frenkel , Phys .Zo it, s .Sow .1etu.nl on 9 533 (1936) (in English) ■- Accord in" to Weisskopf, Phys.Eev. ^2 295, the average energy of evaporated neutron is about 2kT . energy Ch. VIII Nuclear Evaporation 163 energy of the highest occupied state at T - Let i = E - 16 ■ ^, 1G the e xclt -tion ener^. Then the probability that oilfe neutron takes out so much energy that leas than 8 Mcv is left ii " ' "" > - */VT vanes c" d et e rsai ned 8tS 3 8 + 5 c At die = c £, kT where setting P, =r 1 at 6 0; C" p l = - p -feT Probable number evaporated, excitation 8-16 M°v = P + °P = P L + £(1-P 1 ) =.2 - e"^ T 1 For total, excitation 24-32 Mev , there are three eomnetinr process: 1) one neutron loaves not enough energy for further evaporation. P-, = e"^ 8 *»% calculation^'slmilar" to above . This is neplipibl.e. 2) the first neutron evaoorated leaves enough energy for just one further emitted neutron. The nrobabllttv for t'nis is i-tt+s .£. _ S - - - ./ p ? - G l e FT i £ = e * |) the. first, neutron leaves enourh enerpy for two more neutrons, ana either one or tv:o more are omitted. The probability that case 3 J occurs vrith just one neutron beinp evaporated subsequen- tly can he shown to be _„ g / - 1- \ p 2 = .Ff(, e ' V- Then the Probability for ease 3 J with- ftwo subsequent neutrons, i.e., three altogether is ■ j ■ % = 1. - Po = 1 - H - P3 = 1 - The average = 3 - e number for '<>+*) the 24-32 Mev rann.e is The plot turns out as -iven: For protons. the barrier is effectively higher, by an amount that Is mnroximatoly equal to 0,9 times the peck" height of the Coulomb poten- tial. T> Le probability for penetration of a Coulomb barr by a charged particle is P = gl^PT^^'K^ Efret^y infi W)=i O-sy where x = Enerry/Barrier h&i^hi (after Bethe, B*, p. 167) A, and 2 1 are mass and. charge of particle, Z ? is charge giving rise to the barrier. A 2 , the atomic weip-ht, enters through the nuclear r ? ,n;, n . ft -. ™ + „ • Curves of probab- nuclear radius, R^. For a proton, A-.Z-, = 1 ility of oenetration for three valui given: v\ \\ A = 90 \A 2 -a 2 \ X \ A, for protons, are rVotnri rVstnstod/eftcy .7 'Eme rgy « = Barrier Fission Oh. VIII 164 The ratio of probabilities for emission of a neutron and. of a proton is ._. ->■ ?wev C-o on -cm") -4e LtffMtV ~ .00 37 where 9 Mev - 0.9 of the 10 Mev Gculomb barrier. 1.65 is kT c o r re sp o nd ing to an excitation of 38 Kev , av e ra ge numb e r of neutrons emitted = 4. h Fission ** The most useful model for explaining fission phenomena Is the liquid drop model (see chapter 1,0, p. <S ). This model per- mits calculation of the change in potential energy when the nuc- lear drop suffers an ellipsoidal deformation from spherical shape. If the potential energy Increases, spherical shape is a stable configuration. The two contributions to the potential energy are l) capillary energy, 2) electrostatic energy. We wl.ll calculate the change in these contributions to the potential energy for a constant volume prolate ellipsoidal deformation given by the equations ma lor semi-axis = a = R0+ e ) H" minor semi- axis b = R/\fi 1 +£ O— <3P> VI11 - 58 where R = Initial radius, £ = parameter giving extent of deform- ation. (3-S- approaches the square of the eccentricity of the elliptical section as both arra roach 0.) Volume is invariant: V = "(W3)ab 2 _ (4n/3)R3. l) The capillary energy is proportional to the surface area. Ellipsoidal surface = 4tt P 2 ( 1 + 2/5 £*+...) VIII. 59 The capillary energy was c omp.it ed In Ch . I , p . 7 , and found to ,;De 0.014 A^ for an unexcited (spherical) nucleus, therefore capillary merry = 0.014 A 3 (l + 2/5 E*+...) (mass units) VIII. 60 -, n a 2) The electrostatic energy = (3/5) fe-Z^/R) (l - 1/5 %hk &t sphericity the energy is, from Ch . I , p . S , 0.00062 7 Z-/A, therefore Electrostatic energy - 0.000627(Z'-/jml - l/g E ) VIH.61 This is evidently maximum at spherici" The total change is £ (2/5 x 0.014 A 3 - 1/5 0.000627 &'/$%) VIII.6J Spherical shaoe is stable if this is +; unstable, if -. Roughly, ■electrical energy is proportional to A* 3 , capillary energy to A ^, Therefore the electrostatic energy term dominates at high A. The expression VIII, 62 gives a criterion for stability for given Z and A, namely, Cvt-'i - Spherical nuclear drop is stable if t "C4.7 VIII. 63 * Bohr and Wheeler, Phys.Rev. 56 426 (1939); Frenhel , J.oi Physics, A^ad.S c l.U.S.S.R. Vol. 1 Jo.2 (1939) (in English) **Gencral reference: W-.E .Stephen, "Nuclear Fission and Atomic Pov, r er," Lancaster, 1949. Oh. VIII Fission The plot, of Z 2 ,/A shov:s that elements up to U 2 3S stable by t]iis criterion; they are In reality. Although spherical shape of the nuclear drop may be a relative minimum of potential -2 energy (i.e., metastasis), the ^ potential energy may be even louer for completely" separated halves of the drop. To investi- gate this, v:e can compute the enerry of tuo separated spherical drops of equal volume having a total volume equal to that of the combined sphere, and compare this ulth the energy of the original d ro p . The ma s s f o rmu la 1.8, p, 7 , may be used for this ■■ur- pose. If M(Z,A) denotes the mass are Interested in the difference 165 at least, are 50- *■!> — - - -^ 40- 3S- 3<-sS 20- to- /2S D . Z=o 2o 4o SO SO I DO 120 of combined nucleus, ther: ue M(A,Z) HV 2'1 J % 0.014 A 5 (l-2 3 ) + 0.000627 S* (i This gives the volurr. e fragment a Ccrence In energy I or At, closer distances, Infinitely separated half- still not" touching each In FIG. VIII. 7, the other, the 'potential increases due to Coulomb repulsion. These facts permit d raving an approximate potential energy curve. For y236 the separated halves have energy -169 Mev relative to the combined non- excited nucleus, FIG. VI II. 7. early part of the curve represents deformation of the sphere into a prolate ellipsoid of small eccentricity, and Is known from equation Fill .62; it la quadratic in the parameter £ . For Fully seoarated fragments , " ~ no"' the curve Is I) r it to r c , the separation lny^erbolcLj CoulomD betential Distance j mengm^ ith E. at sb^Vi t d^Tov-^dtiDn distance at which the tv.'o half-volume fragments just touch, the potential dev- iates from the Coulomb lau due to the onset of nuclear forces. (in the case of -,f- ^Volume Spheres \S- J0 , if the Coulomb lav; just touching pi S .y[if. f 7 held right un to r , and If the fragments remained strictly spher- ical up "to this point, the potential energy would be 210 Mev J The c v rv e in the re pi on near r is c ompl 1 c a t ed , Cal c : : 1 a t i o n s o f potential energy for large deformations (up to about point B on FIG. VI 1 1. 7) of the nucleus have been performed by Fran!-: el and Metropolis ( Phus.Rev. 72 914 (194?)) using the "Enlac" computer. Due to aero point energy characteristic of systems, the system has energy slightly above t" potential well . There is some chance that undergo " snor-tar.eous" fission the nucleus will :y tunneling qu a ntu m- m e c ha 1 , n I c a 1 ■ e bottom of the fig, vm,7 166 Fission Ch. VIII through the barrier. The probability is lovr, because the masses are large and the system approaches being classical so far as tunneling through barriers is concerned. Therefore, effectively, there is a threshold excitation for fission. In 1939 Bohr ana V.heeler* deduced a rule that has been verified experimentally. It is that the reaction (n,f) 1b pro- duced In elements haying an odd number of neutrons oy thermal neutrons, but with elements having an even number of neutrons, fission Is induced only by high energy neutrons. This is because changing from an odd number of neutrons to an even number of neut- rons releases one or two Mev . This rude agrees with the table of which is reeroduced here. The part Possible Bl'vichviq £^e.v guen aid N Q e o - 7 ~ 5" ~5~ given In Ch. I, p. 7 rule Is exemplified by fr 35 and U 238 . The bind- ing energy of a neutron to the former may be some- thing like 7 Kev; to the 1 a 1 1 er , wh i c h already has an ev e n numb e r of neutrons , only about 5 Mev. Fission induced by thermal neut- rons occurs in Ij^3?5 o U t not in 02^8 _ Xli tnis case, the barrier to fission is presumably around 6 Mev. in height. Asymmetry of Fission Fr , _ to split into equal fragments Ti sent distribution. Nuclei tend not Fragments tend to cluster in two zones, as shown in FIG-. VIII .8 and 9'. Referring to JFIG-.VIII .3 , the fragments, after formation, move toward the line of max- imum stability hy p decay. There is no adequate theory for this a symme t ry . Extensive data or nuc- lear species produced in fis- sion appeared In Rev ,yod . P^s , 18 513" (1946). fission fragments tend to be in these regions, rather than midway . Stable nuclei — i FIG-. VIII, 8 Fission Yield 1 fo I lo" 1 /^il / 140 N^ f\ I \ &o to 1 0<j 1£B l4o L4& ltt.-A Kejnarhs on Fission Fragments . Fission fragments nrroduce heavy cloud chamber tracks about 2 cm. long in air at NTP. * In contrast to ionization produced by a particles, the density of ionization here decreases with d 1 st anee t rav e 1 1 eel , as sho we i n FIG- . V 1 1 1 . 10 . The explanation is that the fragment , Initially to a large extent stripped of electrons, gains electrons as it slows down. Its * Bohr and bheeler, Rhys. Rev. 56 426 (1939) eh. viii Nuclear Orbit Model 167 ■■fission fragment About 1% effective Z decreases, and ionization depends on Z (equation 11.10, 1. *bh (The a picks up char G e also, of course, but the effect Is smaller.) Emission of Neutrons in Fission. Immediately after fission, when the f ramnent" Tr drops' r are some- what like: OQ ' each fi*^>»ttb uossessss considerable excitation energy. This energy la used to a large, extent to evaporate neutrons There is approximately ono neutron produced per fragment Delaved Neutrons of the neutrons emitted by fission fragments are emitted at relatively Ion": times after fission, I.e., from 55 seconds to a fraction of a second. The explanation Is as^ f n m n vrq in the "B decay by which. n f fsio">ragn,ents\ecome stable nuclei, It ^JJWJJ.fjJ § + 1 aeeav from a neutron- rich nucleus 2 to the ground state ot-4 + 1 f'SrMdden. Then the nucleus Z+l is PJ^^^ft 11 ^ , U-+-* n - n fl n aY if It has sufficient energy, transiu^u v ' ^f by^vSorating a" neutron. The neutron binding energy Is relatively small for these neutron- rich nuclei. It can, ol processes leading to the nucleus (Z+1J . aenDLeB e^v Triple Fission. Fission of a large nucleus Into three fragments--- two major fragments plus one alpha particle— Is known. Distance FIG. VIII. 10 Problem. Draw quantitatively as well as possible the curve FIG VIII .7, the energy of a nucleus as a function of some para- meter giving the extent of deformation. Invent a suitable nnwunp+PT to measure distortion. „^„„„ f It is probably simplest to describe the deformation by surface harmonics, I.e. , p _ a + 2> n S n 1 + ■ „ vM + in^ raflltis and Sl, are surface harmonics; for where a is the initial radius ana ^ invarlance Is thus auto- SfcaliroroJldeg rorT Good Approximation to experiment Is obtained by Sling r = a + b 9 P 2 (cos6). The capillary energy is -ents are not joined, but are still close together, tney are deformed something like: QO The energy can be .rudely approximated by assuming each to be an ellipsoid The papers of Bohr and feeler, l,c . , Mel and Met nop oils 1 .c . «* , of Frenkel, J. of Phys. Aca d. Scl .USSR 1, So. 2 (1939) Un *n S _i^- J are pertinent to this problem. ^ E rb 1 1 Model of the Nucleus H^gi 9 |SiSr^ nucleus In terms of nucleon orbits ^J nv°» the ascription of the atom in terms of electron Sits, rnrorbi/olcture'ls valid If collisions are — -ough so that a nucleon may travel at least across the nuel •££**£* col lision. This requirement seems at first not to dc -u±±-l.l_ h;-As far as terms linear in the br J n 168 In a v ucleus , Tor is of the ord er o leons, the saean f the nucleus . How ignore s , and th e s 1 ) When one Nuclear Orbit Model Ch. VIII at ?** 20 Kev the n-p scattering cross- sect ion f 0.3 barn, and, for the known density of nuc- ree path is only about l/j or so the radius of ever, there are two factors which this calculation e make the orbit picture appear not so untenable, ential well. If Spaced hue leons , blend together to 2 ) The uucle energy states are occur between nuc ferring both the principle prevent state . Diagramma nuc 1 eon passes the nucleon is the wells may form a roughl us is a degene , for the most leons only if nucleons to em s two nucleons tlcally , another, It passes through a pot- constantly passing other closely be so closely spaced so as to j uniform potential. rate system in which the lowest part," filled, A collision can the collision results in trans- pty states. The Paul! exclusion of the same kind in the same Initial momenta of two nucleons plotted In mo entum space approximate be few, but some collision cause direction. Bot must represent states. If the states on the 1 lision does not if occupied sta completely fill there could be all. mdary of filled unfilled states _, to change re 1 ew end points prev 1 on sly empty re are no empty ocus A, the col- occur. Thus, tes described a ed perfect sphere, no collisions at state region; here Neither of these tuo Ideas extent that they represent is justified. The orbit model has been explored absence of accurate information, a squ with rounded corners, is adooted as th The depth of the potential is assumed A. This Is justified by the computati showing that the kinetic energy of the ^A*a , is, to fair approximation, inde the gas model. The requirement that t neutron be about 8 Mev then fixes the las been fully investigated . To the -.he true situation, the orbit model ith some success. In the are well, or a square well e form of the potential, not to change much with on at equation VIII. 48 highest occupied state, pendent of A, according to he binding energy for a total depth of the well. Quantum mechanical calculations for the square well give levels which may be denoted as follov. r s : Is Ip Id 2s If 2p lg 2d 3s Ih 2f 3p 11 2g 3d 4s 6 10 2 14- £ \1 id 2 2 2 (4 6: 2£> IS l° Z 2. 2 4D TO \\z 16 8 The letter gives. X in the usual way, that Is, means ^ ~ ~5 , etc. The number gives the nu: s r o. state capacity summed no . of occupants means^r = , f radial nodes. "l" means no node, "2" means one node, etc. (This differs from atomic spectra notation, n - J - 1. ) :ere the number of radial nodes is For a sharp cornered well , the states listed are about equal!] spaced In energy. When the corners Eire rounded off, the states shift so as to clump into the groups of states given in the list Ch. VIII Nuclear Orbit Model 169 above*,* for example, Id shifts closer to 2s. As mentiored above, the depth of the well 1b approximately constant throughout the periodic table The radius R increases With A according to R = 1.5 X 10~H A /3 . For small radius, only the Is states are hound. As the radius Increases, hound states of higher It become possible, in the order listed above. The Paull exclusion principle prescribes the maximum number of one" kind of uarticle that. can occupy a particular space state. just as in electron configuration of atoms. There may be 2(2^+1) identical earticles in the state having angular momentum A . The factor 2 represents the two possible spin orientations. We may attempt to build up the species of nucleus out of nuclear shells, just as the periodic table is obtained in atomic theory: P resumed configuration (P;K) il H 2 D (Is- Is) meaning a nroton in Is, a neutron in Is. Could ' ' ' have spin or 1 and still accord with Pauli nrir.cinle. Actually observed spin 1. 2) He 3 (Is 2 ; is) Spin 1/3 necessanily ; must be opposed. since the proton spins (lsjls"-) Spin l/2 necessarily, similar to above. (is 2 ; Is 2 ) Spin 0, a "closed shell" nucleus. 3) lh> 4.) He The is orbit for both neutrons and protons is now filled, The next orbit, lp, accomodates six neutrons and six protons, Giving 1 " as the next closed shell nncleus. 5) He 5 does not exist; evidently in He 5 , (is ;ls"lp), the It? state is not bound." Hq5 is the compound nucleus * This may be understood by the following argument . Consiaer a sharp cornered square well having ei gen state functions Y ± : lhe perturbation In energy of state k when the corners of the well are rounded can be found "from first order perturbation theory, ««■«£ J- as the unperturbed states, shift in energy is The ■potent to. 1 . = cJj4^M^> .c = fir, J2 joe-rtuvb'.ng" H where ix* is the radial part oif-vf* , H Is the perturbation of the poten tial that occurs when the corners are rounded . Let H = H-, + &* . H 1 gives the rounding on the inside, at. A, H 9 gives the rounding outside, i p r m at_B. „ j& H 2 Is negative, but c| Kl^ > C L \AA*& ] therefore AE is +. Now the reason that states cf higher angular momentum have larger AE's Is that particles in these states spend relatively more time near the edge of the well, in region A. 170 Nuclear Orbit Model Ch. VIII for the reaction n + He 4 — *-jj + He 4 . A scattering resonance Is observed at about 1 Mev , presumably representing the lp level of KrP . In fact, it 'is The sb claimed that two peaks are observed. p l/2 and %/g would be the p., /n and p^ ^ states; an He (Hall and Koonta, Ph . Rev , 7k 6) ti l i 2 2 (la ,lp-;ls ,lp) presumably _E!SrgV of n ; 196) The end this "period" is There are two independent systems., each consisting of a soln 1 (orbital) and of a spin l/2 (intrinsic spin). There are many possible total angu- lar momenta, including the observed, namely l . 16 the next closed shell 31! 3. It has spin and is unusually stable On the orbit model as given so far, the closed spoils for either the proton or neutron confl durations are at 2, 8, 20, 40, 70, 112, 166,.... neutrons and/or protons. For l&vr.e nuclei the Coulomb energy makes the number of proteins less than' the number of neutrons, and the nucleus cannot have closed shells of both at the same time . VIII. 64 of neutrons, Empirically, the closing numbers appear to 2, 8, 20, 50, 82, 126 (Magic numbers) Nuclei having either such a number of protons, c are unusually stable. These are not the same as the closed shell numbers given bv the preceding development. This discrepancy has been Interpreted by M. a. Mayer as follows. Suppose that spin-orbit coupling splits the energy levels corresponding to different J values, that is, lg splits into lSoy 2 and l®^'. Assume that the level with larger J is mono stable, i.e. lies lower. This assumption is not contrary to any known facts about the nucleus. Then the former closed shell number 40, for example, must be alt ©red as follows: Old dividing line between shells (Higher i numbers are a a erased to have larger Bplittmg) 5~0 ]nuc|eans u|» to here Hew dividing line between shells M. d. Mayer, Phys.Rev . 7_4 235 (1948) Ch. VIII Radiative Capture H(n,"tf )D The next shells are 1&7/2 2^ 3s lh, 171 T.l/2 1 82 nucleons to here lh g/2 2£ 3P lt 13 / 2 , 126 nucleons to here Thus the altered nuclear shell theory Gives closed shell numbers that anree with the numbers deduced from experiment. Problem. Look up the nuclei having; closed shells of eitner neutrons on protons, and note to what extent they tend to toe relatively stable . L. Capture of Slov.- Keutrons by Hydro gen This nu clean reaction is one of the few that can be calcu- lated with some precision. Only the S wave comoonent of the resolution of the neutron plane wave into radial functions is important for the low neutron enerrj.es to be considered here, i.e., thermal energies, ^ > O waves have angular m omen turn with respect to the proton so great as to keen a slov: neutron beyond the reach of nu clean forces. (See chanter VI, section c, , p.ue-rtj The only source of angu- lar momentum is intrinsic spin. The final state Is the deuteron 3^ state (the 1 S state Is not bound). n(slow) + H ; — >• D + a ( itiw = The conceivable transitions are 1 .19 Mev, "S (continuum) ■^S (continuum) ( magnetic dipo'Ls)^ 3g ■ ->- 3 S (is not possible, a s shown just after eq. VI 1 1. 78) The parity selection rale excludes electric dipole for S-> S transitions, Ch. V, p. loo . The physical mechanism, in classical terms, is the following. Since the magnetic moments of the proton and neut- ron are not equal , ^p ~P J^h j the totaly^f for the deuteron does not have the sane direc- tion as the annular momentum vector, FIG. VI II. 11. There- fore the .system is a rotating magnetic dipole. The formula for magnetic dipole radiation is almost identical to that for electric dipole radiation, differing only In that the magnetic "moment matrix element Is used (see Ch . V, p. 95 ) Momenta 2.7 Moment* j ^ Y-W resultant magnetic moment moves around the fixed I axis FIG-. VIII. 11 178 Ch. VIII Radiative Capture H(n, t )D Probability of transition ^ ^ \ u, \ t,,,~r ,-- 7 unit time =^X^S l/NWl.Jwfad I VIII. 65 oer where W = (energy of transition)/^.. The energy of the transition is 2.2.3 Mev, the binding energy of the deuteron, provided the energy of the incident neutron is sraall . The wave functions .are [final) T(%) = (++) ., j( (+-) + (-+) )l/j? s z s a = + i (initial) % } s ^V( + _)^-+3]% yiIIt66 Notation for the spin functions: (+-) siear.s proton spin up,(+)j neutron spin down, (- ) . The space parts of the wave functions are independent of angle, being S functions. u(r)/r is the solution of the Schrodinger equation for the known potential, U-* , the triplet S state neutron-proton potential The radial equation is M-U) + T^(^)(\W 3 -U 3 !^)) J W = VIII. 67 W-, = -2.23 Mev; U4 - -21 Mev for r smaller than the range of nuc- lear forces, and beyond. (See theory of the deuteron in Ch.vi , p. us .) u(r) has the form: u(r) c£ © ' for r>R. k* - \ YmV#J - 2.26 x 10 12 cm" 1 ** -hit The error committed in using u(r) = G g 3 for all r is tolerable, since R < l/k v and we shall correct partly; as follows. When u(r) is set — C, 6" 5 , we get somewhat too large values in integ- rations over r. If we compute the normalization C using the same approximation, an error is made which tends to compensate. Normal- izing so that , f * ..j fir = 4TC Me 5 | d& = 1 we get So VIII .68 VI II. 69 VIII .70 j(r) is the S wave component of the resolution of a plane wave Q-~* s into polar eigenf unctions, but perturbed by the poten- tial well of the proton, The potential well is U-, , the potential existing between a neutron and a proton in a sir. "let S state. Q* & is already normalized to one particle per u""it volume in the absence of the 'oertnrbinr. ootential well, The unperturbed plane wave can be exuanded as follows* &K IM^)/&)"LN Pit— ) VIII .71 *3tratton, "Electromagnetic Theory," p. 408 Oh. VIII The S wave Is Radiative Capture H(n, T )D WfcS, -) 4N JkJt> 173 VIII. 72 We must now find how U perturbs this. The m mature of the wave function is areat+y incr eased in the region of the well and is nearly Independent of neutron enor-y, "since V-c neutron energy is small compared to the well de^th". From i'ib theory of scattering of neutrons by nrotons, Ch . VI, m Itnow t'^t for si ok neutrons the tangent to the radial function ) (r) at the noter,- tial well ed-e V, intersects the axis at a distance a-, from r - Therefore ;.{r) must have the form: W =j(x) i=*r- tje^mntn^ o"f tne function — - in fe vm.73 It Is evident franj the fi-are that j ( r )« (l/h) sin(kr + ia.,0 a fairly good approximation to the tru for small r. Computation of the matrix element t * ! fimt j (r) . j {r}^ JajJ + r The matrix element is VIII. 74- The marnetic moment vector is VI I I. 7? where C T etc . are the Paull spin operators for proton and neutror These operators act as follows: <X x (+)^(~) a(+)^i(-) cr„(+) — (+) i 6 VIII. 7G For example, C^ ( + _ ) _ („) f le£vlnr> thG lzeutTOn sr , ir , B ymtol u^- cna n red . We c al c a 1 a t e (jj^) a s a n ex ampl e . For this we need J^<|^ +ja n CT 71 ) , the spin parts of which -ive r nw — / rr^ 3 ^ ^ / \7. Therefore (Note t hat if Jj F - ^J N the matrix el em ent would be 0. H>- sleally this corresponds to the case that the resultant ma -net! c moment has tne same direction as the an^lar momentum, Which is invariable 1V4 ' — -— Ba&latlye Capture H(n, 5 )D Ch. VIII and the average magnetic moment would be constant,) The term above has the same form as the spin function for the final 3 Z - state, and is multiplied by C++) 3-*-M i> = i^ [c+-)+£-fn^ s,-o Due to orthogonality of the spin functions, only the ¥ S z = gives a contribution. Therefore final for reduces Therefore VIII. 78 It can be seen now that -^S (continuum) ''S (bour.d) does not occur, for in such a transition the matrix element depends on the integral over two S wave functions for the same potential, and this is 0, by orthogonality of eigenfunctions of the same Hamiltonian. In the transition considered above, 1 S — 1-^3, the 3 functions are for different potential functions. Since there is no preferred axis, \(df.J)^\ 2 =^f./h\ l ~\^li)iJ\ \M U V=3Wi,)S TOI - 79 from VIII. 65, .From the definition of the cross section & , a we get °C^r) i#r-rAd V ji*}Mfifa VIII .80 where a factor l/4 comes from the fact that only l/4 of the ini- tial continuum states of the neutron and proton are singlet S. A Using the values: I13 = 2.26 x 10 12 OL a l CA K 10 2.32 x 10" 12 cm cm W rus VI II. 81 VIII .82 §W velocity (f%j For thermal neutrons, V = 2 ..2 x 10 5 cm/ sec, 0^*} = 0.29 barn. The experimental value is O.30 barn. An elaborate theory includ- ing tensor forces gives 0,31 barn. Remark s on neutr on canture in 1 i ght nuclei. C^caVM is in the range .001- .01 barn, with some exceptions, H, with c^ feaj^tj = »% the Ch. VIII Photonuclear Reactions 175 jvL°£2 f + them - p is rela tivei y meb a ls due to fche fact that :!.:^ dCU !'f ro r- State is alcost bo1 -^, i.e., because I aj is exceptionally large light elements, but . There are other cases of large.-*** in the toy the reaction ( n , pa rt i c i e ) : B^O C( n ,a) - 3800 barns He?, 0"(ii,p) = 6000 Li f.3 cr( a , a -*. , - !«.»«, i - 8000 W 14 , C(n,p) = 1.7 + ?\f 0r fI ' J? SfS'll, compared to the others, because, in contest factor ISf^f i ? 8M * 8 ' + V 8 larG6 en ° U ^ t0 Eake *S barrief iactor ejcpt-g-j » important for emission of a charged uarti-cle IftEfSSer'or g*.**"^ ° f * ^on is reduced^ fgcSr" M. Fhotonuclear Reac tions The Be are :nainiy"rY7n), althou^ U.2r_), (J.fj, etc are hnown also. The threshold for (jr.n? is the binding energy If «°° n ^ USUal ? ab ° Ut , 8 Mev " ^eptions te this figra are ■■■"■-■■ : thresholds of 2.2 and I.71 Mev, respectively Strong J absorption resonances have been observed at 30 Mev ( t h re sh o 1 d 18 . 7 Mev } , m ,Gu 2^^l; at 22 Key, and in „^Ta at 16 Mev, Goldhaber and Teller *» have interpreted these res- onances as due to an electric dipole interaction in ,M '\b the neutrons." Btt ° 1 * lB mm a ' S a Unit with ro ^ ect ^ aS the Phot gd i s 1 nt e ,-rs.t io n of the de utero r pliotonuclear reaction having a reasonably The transition if + D — *-n + p (unbound.) This is the only reasonably detailed theory. Except at 1) 2) lov J S 3 Q (33 .ground (D ground fttto p (unbound) occurs in two ways: 3G^r.ej -*- b I continuum) sl c»n e ) -^ s- ^p [ ooati nuum ) er.erraes tenths of an Mev above gular to be - - ---, —.Av^tjiwo-g \Li_.j to a fe 1 .-: tenuns or an threshold), the latter is the dominant reaction. The momentum of the P state requires too high a velocity attained by lov,- energy neutrons. rp , p + ^ fjr* m °f' 3 S— ^(continuum), is the inverse to the reaction discussed in the ore ceding section. Therefor© its cross section is obtainable from detailed balance arguments, allied e #*£ be va^VnTJ^ 6aUatl ° n VIII - Bl ls ^ st *&»*' iUocl t0 bs valid ^or ail neutron energies. The result is S .U - Vi* ^ ^ = H' EK + W ' f,(j +W.) VIII.83 where Cu D is tbe angular frecuercv at threshold, namrdv finding energy W, and Wo is the magnitude of the fictitious binding, energy of the deuteron sin -let state **** " " " * For definition of & see VII 1, 12 or I II. 3, o 58 ^ Goldhaber and Teller, Phvs.Rcv. 74 1046 fl948). **"* Bethe , D , p . 58 " " ***•* W Q is really a parameter such that state were bound, its energy would have to be W tne observed singlet scattering cross section, FIGURE in this footnote, which is continued on if the singlet deuteron in order to Shown on ti- the next page p jivo 176 Photouuclear Reactions Gh, VIII The cross section for the second tvne of transition 3s-»-3p can he computed usiri" the formula for G amma ray c^sorptiiii.:* CT^^^lMp VIII. 84 ^a la the reduced mass, ano v the velocitv of the emitted particle. M is the matrix element for the electric dipale moment M -sr«* i »'^8* VIII .85 £»« ^ coordinate of the proton with resect to the cento- of ST+H^ 6 + faCt ? r tf? rssulta because the neutron is uncharged, ano tiie system is lihe half a dipole . Wis will not carry out the calculation, taut merely describe the approximate wave functions. Except when hi rh est accuracy is wanted, T, 3 „. may he featoan (fro m e qua 1 1 o u VI 1 1 . 7 ) . In t-h e absence of the umlear got ©Btlal *(3p) is ths P wave component of a plane wave (see equation VI II. 71), and has the form I {$*** ^i^ ^^ + , „ .SaITCl &&-#*-} iron tne properties ^ x ^ a -^-°Et d Vessel functions.** At loir energies, (A) ir FIG. vi_i.it;, Tp p j is nearly in the reel on of the short range nuclear potential well-, and Is therefore little disturbed by it, una is like tne P wave of a free particle. At hlrdier ener- gies ¥ ( 3 p ) lias shorter period, and is Perturbed m 0re by the wall (B) la FIG. VIII.12. *fe PIG. VIII.12 r¥ is plotted in FIG. VIII.12 because the behavior of I is seen best when matrices are written in the form, ^cc Unf*) f (vif;) ch. The result is: ^ U CS» f 7 3 M cOj d J Uj 2 - j VIII .86 where UJ 6 is the frequency at threshold, as before. The two cross sections e^Oiwfctffc) and ^^(etetii^are plotted In FIG. VIII. 13 fro**.?. ^ " actual * Pethe D, ■ ** Schiff, n". — a, . 56 77 fictitious Uond state T^ _^ Wing same Qi j nud eneig/ Wo. Very High Energy Reactions 177 e\ectvic of If The reason that Cfy^fw^) rises so sliarply is that energy above the threshold (0 excitation) is almost a resonance energy. Ac can be seen from equation VIII .83,, if W q were zero, there would be a resonance at excitation energy. g. Remarks or Very Hirfi Enerr.Y Phenomena - a) Stars . A rroninent feature of very high energy phenom- ena is the production of stars. These are seen in photographic emulsions and in cloud chambers. The prongs are due to ionizing ^articles, protons, albhas, or larger fragments. Neutrons leave no trace. The quantitative interpretation is rudimentary. Star production car be interpreted roughly as evaporation of particles from a very high temperature nucleus. M. Goldberger*- has added to the evaporation model consideration of the situation when a very high energy nucleon, of energy around 100 Mev, has first entered the nucleus. At this tine, when its wavelengh is very email and its energy is still undistributed among the other nucleons, it has collisions which knock other nucleons out of the nuc- leus immediately. Soon, however, the energy not curried off by these quickly escaping nucleons is distributed in the form of statistical excitation of the nucleus. Then the escape of nucleons is described by the slower arocess of evaporation. Goldberger calculates that about 1/2 of the initial 100 Mev leaves immediately t and about 1/2 is trapped and produces evaporation. b) Deuteron st r 1 y, ping ■ ** Deuterons, given an energy of 200 Mev In the Berhelev 184" cyclotron, impinge on a target Inside the cyclotron, and" a beam of very high energy neutrons emerges. (FI&. VII I. 14 J The interpretation is that the proton of the loosely bound deuteron hits and is caught by a nucleus, and the neutron flies on. Ideally, the neutron would have about 1/2 the energy of the deuteron. However the neutron has a velocity with resnect to the center of mass of the deuteron. After the proton is removed, the neutron's total velocity vector is the vector sum of the velocity of the center of ma s s of the d eu t e r on and th e v el - ocitv of the neutron relative to the center of mass at the Instant the proton is removed. Therefore the neutrons emerge with a spread in energy and in angle . StaP * M.L. Goldberger, Phys.Rev . Jh 1269 (1948) ** Helrr.olz, McMillan and Sewell, Phy s . Rev . jS 1003 (1947) 178 Very High Energy Phenomena Ch. YIII >m tyclotrcn tjeuteron ^am CYe.UtVW (of at on oeam FIG. VIII. 14 FIG. VI II. 15 e;) b s e rv ocl exchange 1 ri p- n 3 c a 1 1 e r i n g . Wh e r. a be am of 350 Mey protona hits a scatterer placed, inside the Berkeley cyclo- tron, FIG. VI II. 15 , high energy neutrons (as well as protons) emerge. This shov.'s that in many collisions the neutron and proton exchange roles: 3ccttie rer ^ ^__ . ^- R^jftyon A similar effect has been observed in ft-.-rj scattering.'* Problem: Compute the distribution in energy and angle of scat- tering of neutrons re suiting from the stripping of fast deuter- ons , using the simple mod~l described above. (This calculation Is done in the first part of Berber 1 s paper on the theory of stripping of deuterons, Phy a ■ Rev . 7_2 1007 (l94y). Although the most probable neutron energy is l/2 that of the deuteron, the energy spread, is larger than the binding energy of the deuteron by a factor of (l/2 E d /EE(d) ) /i «{ 100/2 y z ar 7. This may seem like violation of cor.sen.rat ion of energy, Houever, the velocity of the neutron relative to the laboratory system = '^^_ elx ^ t + ^ rel _ c i eu t. > and - t - le elle - T CJ °f the neutron In the lab. system thereforeis ^M(aij 1 + fe.^ifed^O^ + Zarj^^.fad-t) :ere the last term, is the marl contribution to the snread, For a su mm a ry of very hi gh e ne r f-y phenomena, s e i Chew and Moyer, Are. J. Physics IS 1S5-1J5, Ig 17, 22 20 5- *This is discussed further under "Exchange Forces" in Ch. VI, p. 122